Period comparisons

mark

Answer the following question, where the numbers [imath]i[/imath] and [imath]j[/imath] are given below:
Suppose [imath]f:\mathbb R \to \mathbb R[/imath] is continuous and has a point of period [imath]i[/imath]. Must it it have a point of period [imath]j[/imath]?

a. [imath]i = 56[/imath] and [imath]j = 104[/imath]
b. [imath]i = 48[/imath] and [imath]j = 104[/imath]

eric

A) 56 and 104 falls in the 3rd tier of the Sarkovskii ordering ie , (2n+1) (2^3) ; {....56,72,88,104.....} so 56 < 104. meaning period 56 also has period 104.
b) From part A we know 104 falls in the 3rd tier of the Sarkovskii ordering ie (2n+1) (2^3) ; Now 48 is actually the first number on the 4th tier ie, (2n+1)*(2^4); {48, 80, 112, 144, 176....}

Table[(2 x + 1), {x, 1, 20}]
Table[(2 x + 1) 2, {x, 1, 20}]
Table[(2 x + 1) 4, {x, 1, 20}]
Table[(2 x + 1) 8, {x, 1, 20}]
Table[(2 x + 1) 16, {x, 1, 20}]

eric

Tier 1; {3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, \
39, 41}
Tier 2; {6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, \
74, 78, 82}
Tier 3; {12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, \
140, 148, 156, 164}
Tier 4; {24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184, 200, 216, 232, \
248, 264, 280, 296, 312, 328}
Tier 5; {48, 80, 112, 144, 176, 208, 240, 272, 304, 336, 368, 400, 432, 464, \
496, 528, 560, 592, 624, 656}